1. Field of the Invention
The present invention pertains to a method for manufacturing a semiconductor device, a method for efficiently simulating and analyzing the structure of a semiconductor device during a device designing stage, a semiconductor device analyzer for carrying out such a method, and a storage medium storing a program to execute such a method. In particular, the present invention relates to a method for efficiently analyzing the influence of parasitic elements embraced in a semiconductor substrate on which fine circuit patterns are formed, an analyzer for executing such a method, and a storage medium storing a program to execute such a method.
2. Description of the Related Art
Semiconductor devices with high-frequency analog circuits, analog-digital mixed circuits, and high-speed digital circuits have fine circuit patterns delineated by, for example, photolithography on semiconductor substrates made of, for example, silicon (Si) and gallium arsenide (GaAs). Testing and evaluating these complicated and advanced semiconductor devices need time and money if they must actually be fabricated for each test or evaluation. To minimize such time and money, it is essential to analyze and verify the physical and electric characteristics of a semiconductor device before actual fabrication thereof. The semiconductor devices with high-frequency analog circuits, analog-digital mixed circuits, and high-speed digital circuits involve a problem that signals transmitting through a semiconductor substrate interfere with circuit elements merged in the substrate. The designing of such semiconductor devices, therefore, must be made by correctly and speedily analyzing the influence of parasitic elements encompassed in semiconductor substrates.
Analyzing a semiconductor device involves errors, and if the errors are large, trouble will arise during the manufacturing of the semiconductor device. The errors must be minimized to reduce loss in fabrication of a prototype semiconductor device that takes several weeks of intricate processes. If the prototype semiconductor device shows unacceptable behaviors, it must be redesigned and reproduced through the long and intricate processes, to waste time and money. Improving the correctness of semiconductor device analysis, therefore, is critical to shorten the development and manufacturing periods of semiconductor devices. There is no way to restore time loss caused by design failure. Semiconductor industries are competing for higher-performance semiconductor devices, and the most critical issue for them is a development speed. Design and development periods must be shortened.
B. R. Stanisic et al. describe the effectiveness of modeling a semiconductor substrate as a three-dimensional resistive network and analyzing the resistive network with a circuit simulator (IEEE Journal of Solid-State Circuits, vol 29, No. 3, pp. 226–238, March 1994). The correctness of the resistive network model of this technique depends on the fineness of three-dimensional meshes that form the resistive network. If finer meshes are defined to improve the correctness of the resistive network, even the substrate part of a given semiconductor device will involve a large-scale network to take a simulation time longer than an actual fabrication time, or in some case, the analysis itself will fail.
To cope with this problem, there is a technique of reducing the scale of a substrate network model. After forming a resistive network model for a semiconductor substrate, this technique eliminates nodes that are not directly connected to circuit elements such as transistors, power supply wirings, and ground level contacts. The resistive network model or substrate network model is a collection of three-dimensional meshes or unit cubes. Each apex of each unit cube defines a node. The unit cubes form a large cube, and nodes that are contained under the surface of the large cube are “internal nodes” and nodes that are at the surface of the large cube are “external nodes.”
To reduce the scale of a substrate network model, an admittance matrix (consisting of Y-parameters) is prepared for the substrate network model, and through matrix operation, a smaller-scale equivalent matrix is made. The dimension of the equivalent matrix is equal to the number of external nodes of the substrate network model. Assuming that the substrate network model has n nodes in total with m nodes being external nodes connected to surface device/wiring regions and “n−m” nodes being internal nodes contained in the substrate, the admittance matrix (hereinafter referred as “the Y-matrix”) for the substrate network model is as follows:                     Y        =                  [                                                                      Y                  11                                                            ⋯                                                              Y                                      1                    ⁢                    m                                                                                                Y                                                            1                      ⁢                      m                                        +                    1                                                                              ⋯                                                              Y                                      1                    ⁢                    n                                                                                                      ⋮                                                                                                                          ⋮                                            ⋮                                                                                                                          ⋮                                                                                      Y                  m1                                                            ⋯                                                              Y                  mm                                                                              Y                                      nm                    +                    1                                                                              ⋯                                                              Y                  mn                                                                                                      Y                                      m                    +                    11                                                                              ⋯                                                              Y                                      m                    +                                          1                      ⁢                      m                                                                                                                    Y                                      m                    +                                          1                      ⁢                      m                                        +                    1                                                                              ⋯                                                              Y                                      m                    +                                          1                      ⁢                      n                                                                                                                          ⋮                                                                                                                          ⋮                                            ⋮                                                                                                                          ⋮                                                                                      Y                  n1                                                            ⋯                                                              Y                  nm                                                                              Y                                      nm                    +                    1                                                                              ⋯                                                              Y                  nm                                                              ]                                    (        1        )            
The Y-matrix (1) contains the following sub matrices:                     A        =                  [                                                                      Y                  11                                                            ⋯                                                              Y                                      1                    ⁢                    m                                                                                                      ⋮                                                                                                                          ⋮                                                                                      Y                  m1                                                            ⋯                                                              Y                  mm                                                              ]                                    (        2        )                                B        =                  [                                                                      Y                                                            1                      ⁢                      m                                        +                    1                                                                              ⋯                                                              Y                                      1                    ⁢                    n                                                                                                      ⋮                                                                                                                          ⋮                                                                                      Y                                      mm                    +                    1                                                                              ⋯                                                              Y                  mn                                                              ]                                    (        3        )                                C        =                  [                                                                      Y                                      m                    +                    11                                                                              ⋯                                                              Y                                      m                    +                                          1                      ⁢                      m                                                                                                                          ⋮                                                                                                                          ⋮                                                                                      Y                  n1                                                            ⋯                                                              Y                  nm                                                              ]                                    (        4        )                                D        =                  [                                                                      Y                                      m                    +                                          1                      ⁢                      m                                        +                    1                                                                              ⋯                                                              Y                                      m                    +                                          1                      ⁢                      n                                                                                                                          ⋮                                                                                                                          ⋮                                                                                      Y                                      nm                    +                    1                                                                              ⋯                                                              Y                  nn                                                              ]                                    (        5        )            
With these submatrices (2) to (5), the Y-matrix (1) is expressed as follows:                     T        =                  [                                                    A                                                                                                                          B                                                                                                                                                                                                                                                                                                                                      C                                                                                                                          D                                              ]                                    (        6        )            
An equivalent m-dimensional matrix Y′ is obtained as follows:Y′=A−BD−1C  (7)
Through these operations, a given substrate network model is reducible to an equivalent matrix of smaller dimension.
This technique, however, is not practical. The reason will be explained in connection with a CMOS integrated circuit. The surface of a semiconductor substrate has impurity-doped regions of different conductivity types. The impurity-doped regions include an n-well for a pMOS transistor and a p-well for an nMOS transistor. A p-n junction boundary between the impurity-doped regions of opposite conductivity types involves junction capacitance and capacitive elements. Accordingly, admittance (y-parameter) between a node “i” in the n-well and a node “j” in the p-well involves a capacitive component. Namely, the admittance of a three-dimensional mesh in the semiconductor substrate of a real semiconductor device is expressed as follows:Yij=gij+jωcij  (8)
Namely, the admittance is dependent on a frequency. In this equation, gij is the transconductance “1/rij” of a resistive element having a resistance of rij [Ω], cij is a capacitance cij [F] of a capacitive element, and ω is an angular frequency. To execute the matrix operations mentioned above by computer, each matrix element must be converted into a numerical value. This is needed for every angular frequency ω to analyze, to extremely increase the number of calculations and deteriorate analytic efficacy.
To avoid this, the above equations may symbolically be processed. Programming the symbolical processes, however, is practically impossible because original equations to be reduced are of large scale and are complicated with several thousands of variables.
To correctly analyze the influence of parasitic elements in the semiconductor substrate due to elements merged in the semiconductor substrate, it is necessary to consider not only resistive elements but also capacitive elements in the substrate. Using resistive and capacitive elements to model the admittance of the substrate increases the number of nodes and must consider the frequency dependence of the admittance. This results in elongating an analysis time and impeding analysis. The earlier methodology, therefore, is unable to correctly analyze the influence of parasitic elements in the semiconductor substrate nor efficiently design the semiconductor device.
The conventional Y-matrix reduction technique of eliminating irrelevant nodes is impractical.